# Can a Quantum Walk Tell Which Is Which?A Study of Quantum Walk-Based Graph Similarity

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## Abstract

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## 1. Introduction

- We give a formal proof of the positive definiteness of the undirected kernel for the case where the Hamiltonian is the graph Laplacian and the starting state satisfies several constraints;
- We propose a simple yet efficient quantum algorithm to compute the kernel;
- We perform an empirical comparison of the performance of the kernel for both directed and undirected graphs and for different choices of the Hamiltonian.

- Adding the edge directionality information allows to better discriminate between the different classes, even when compared with other commonly used kernels for directed graphs;
- In most cases the incorporation of the node-level topological information results in a significant improvement over the performance of the original kernel;
- The optimal Hamiltonian (in terms of classification accuracy) depends on the dataset, as already suggested in [18];
- The constraints we enforce to ensure the positive definiteness of the kernel disrupt the phase of the initial state, leading to a decrease in classification accuracy.

## 2. Graphs and Quantum Walks

#### 2.1. Elementary Graph-Theoretic Concepts

#### 2.2. Quantum Walks on Graphs

#### 2.3. From Quantum Walks to Graph Density Matrices

## 3. Graph Similarity from Quantum Walks

#### 3.1. Extension to Directed Graphs

#### 3.2. Integrating Local Topological Information

#### 3.3. Kernel Properties

**Theorem**

**1.**

**Proof.**

#### 3.4. Kernel Computation

## 4. Experiments

**MUTAG**[40] is a dataset consisting originally of 230 chemical compounds tested for mutagenicity in Salmonella typhimurium [41]. Among the 230 compounds, however, only 188 (125 positive, 63 negative) are considered to be learnable and thus are used in our simulations. The 188 chemical compounds are represented by graphs. The aim is predicting whether each compound possesses mutagenicity.

**PPIs**(Protein-Protein Interaction) is a dataset collecting protein-protein interaction networks related to histidine kinase [42] (40 PPIs from Acidovorax avenae and 46 PPIs from Acidobacteria) [43]. The graphs describe the interaction relationships between histidine kinase in different species of bacteria. Histidine kinase is a key protein in the development of signal transduction. If two proteins have direct (physical) or indirect (functional) association, they are connected by an edge. The original dataset comprises 219 PPIs from 5 different kinds of bacteria with the following evolution order (from older to more recent): Aquifex 4 and Thermotoga 4 PPIs from Aquifex aelicus and Thermotoga maritima, Gram-Positive 52 PPIs from Staphylococcus aureus, Cyanobacteria 73 PPIs from Anabaena variabilis and Proteobacteria 40 PPIs from Acidovorax avenae. There is an additional class (Acidobacteria 46 PPIs) which is more controversial in terms of the bacterial evolution since they were discovered.

**PTC**(Predictive Toxicology Challenge) dataset records the carcinogenicity of several hundred chemical compounds for Male Rats (MR), Female Rats (FR), Male Mice (MM) and Female Mice (FM) [44]. These graphs are very small and sparse. We select the graphs of Male Rats (MR) for evaluation. There are 344 test graphs in the MR class.

**COIL**Columbia Object Image Library consists of 3D objects images of 100 objects [45]. There are 72 images per object taken in order to obtain 72 views from equally spaced viewing directions. For each view a graph was built by triangulating the extracted Harris corner points. In our experiments, we use the gray-scale images of five objects.

**NCI1**The anti-cancer activity prediction dataset consists of undirected graphs representing chemical compounds screened for activity against non-small cell lung cancer lines [46]. Here we use only the connected graphs in the dataset (3530 out of 4110).

**Shock**The Shock dataset consists of graphs from a database of 2D shapes [47]. Each graph is a medial axis-based representation of the differential structure of the boundary of a 2D shape. There are 150 graphs divided into 10 classes, each containing 15 graphs. The original version contains directed trees each with a root node, the undirected version has been created by removing the directionality.

**Alzheimer**The dataset is obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) [48] and concerns interregional connectivity structure for functional magnetic resonance imaging (fMRI) activation networks for normal and Alzheimer subjects. Each image volume is acquired every two seconds with blood oxygenation level dependent signals (BOLD). The fMRI voxels here have been aggregated into larger regions of interest (ROIs). The different ROIs correspond to different anatomical regions of the brain and are assigned anatomical labels to distinguish them. There are 96 anatomical regions in each fMRI image. The correlation between the average time series in different ROIs represents the degree of functional connectivity between regions which are driven by neural activities [49]. Subjects fall into four categories according to their degree of disease severity:

`AD`—full Alzheimer’s (30 subjects),

`LMCI`—Late Mild Cognitive Impairment (34 subjects),

`EMCI`—Early Mild Cognitive Impairment (47 subjects),

`HC`—Normal Healthy Controls (38 subjects). The LMCI subjects are more severely affected and close to full Alzheimerś, while the EMCI subjects are closer to the healthy control group (Normal). A directed graph with 96 nodes is constructed for each patient based on the magnitude of the correlation and the sign of the time-lag between the time-series for different anatomical regions. To model causal interaction among ROIs, the directed graph uses the time lagged cross-correlation coefficients for the average time series for pairs of ROIs. We detect directed edges by finding the time-lag that results in the maximum value of the cross-correlation coefficient. The direction of the edge depends on whether the time lag is positive or negative. We then apply a threshold to the maximum values to retain directed edges with the top 40% of correlation coefficients. This yields a binary directed adjacency matrix for each subject, where the diagonal elements are set to zero. Those ROIs which have missing time series data are discarded. In order to fairly evaluate the influence caused by edges directionality, an undirected copy has been created as well. In particular, let ${A}_{d}$ be the adjacency matrix of a directed graph. Then its projection over the symmetric matrices space will be given by $({A}_{d}+{A}_{d}^{\top})/2$.

#### 4.1. Experimental Framework

#### 4.2. Undirected Graphs

#### 4.3. Directed Graphs

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Given two graphs ${G}_{1}({V}_{1},{E}_{1})$ and ${G}_{2}({V}_{2},{E}_{2})$ we build a new graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ where $\mathcal{V}={V}_{1}\cup {V}_{2}$, $\mathcal{E}={E}_{1}\cup {E}_{2}$ and we add a new edge $(u,v)$ between each pair of nodes $u\in {V}_{1}$ and $v\in {V}_{2}$.

**Figure 3.**The graph obtained by merging two undirected graphs with 2-dimensional node signatures. The tickness of the edges is proportional to the similarity between the signatures of the nodes being connected.

Datasets | MUTAG | PPI | PTC | COIL | NCI1 | SHOCK | ALZ |
---|---|---|---|---|---|---|---|

Max # vertices | 28 | 232 | 109 | 241 | 106 | 33 | 96 |

Min # vertices | 10 | 3 | 2 | 72 | 3 | 4 | 96 |

Avg # vertices | $17.93$ | $109.60$ | $25.56$ | $144.97$ | $29.27$ | $13.16$ | 96 |

# graphs | 188 | 86 | 344 | 360 | 3530 | 150 | 149 |

# classes | 2 | 2 | 2 | 5 | 2 | 10 | 4 |

**Table 2.**Classification accuracy (±standard error) on undirected graph datasets. The best and second best performing kernels are highlighted in bold and italic, respectively.

Kernel | MUTAG | PPI | PTC | NCI1 | COIL |
---|---|---|---|---|---|

QJSD${}_{A}$ | $86.72\pm 0.14$ | $78.71\pm 0.30$ | $56.09\pm 0.15$ | $66.90\pm 0.03$ | $69.90\pm 0.08$ |

QJSD${}_{L}$ | $84.92\pm 0.18$ | $73.79\pm 0.42$ | $\mathbf{59.70}\pm \mathbf{0.16}$ | $\mathit{69.48}\pm \mathit{0.03}$ | $\mathit{70.72}\pm \mathit{0.07}$ |

QJSD${}_{NL}$ | $87.10\pm 0.14$ | $74.63\pm 0.37$ | $55.16\pm 0.18$ | $66.36\pm 0.03$ | $69.61\pm 0.10$ |

QJSD${}_{A}^{hk}$ | $\mathbf{88.51}\pm \mathbf{0.13}$ | $\mathit{81.56}\pm \mathit{0.34}$ | $58.76\pm 0.14$ | $63.88\pm 0.03$ | $69.68\pm 0.06$ |

QJSD${}_{L}^{hk}$ | $86.36\pm 0.16$ | $77.08\pm 0.28$ | $57.63\pm 0.13$ | $64.96\pm 0.02$ | $70.24\pm 0.06$ |

QJSD${}_{NL}^{hk}$ | $\mathit{87.79}\pm \mathit{0.12}$ | $74.38\pm 0.37$ | $58.45\pm 0.16$ | $64.01\pm 0.04$ | $70.55\pm 0.09$ |

QJSD${}_{A}^{wk}$ | $85.97\pm 0.14$ | $74.91\pm 0.32$ | $\mathit{58.91}\pm \mathit{0.13}$ | $63.52\pm 0.05$ | $70.48\pm 0.06$ |

QJSD${}_{L}^{wk}$ | $85.81\pm 0.16$ | $\mathbf{84.66}\pm \mathbf{0.26}$ | $58.01\pm 0.13$ | $64.45\pm 0.03$ | $\mathbf{71.34}\pm \mathbf{0.05}$ |

QJSD${}_{NL}^{wk}$ | $87.61\pm 0.16$ | $74.74\pm 0.30$ | $57.46\pm 0.16$ | $63.34\pm 0.04$ | $70.31\pm 0.06$ |

QJSD${}_{*}$ | $75.47\pm 0.18$ | $56.50\pm 0.41$ | $58.15\pm 0.12$ | $62.89\pm 0.02$ | $13.51\pm 0.16$ |

QJSD${}_{*}^{hk}$ | $70.99\pm 0.12$ | $70.53\pm 0.31$ | $55.29\pm 0.13$ | $67.19\pm 0.01$ | $34.24\pm 0.22$ |

QJSD${}_{*}^{wk}$ | $79.44\pm 0.16$ | $70.21\pm 0.40$ | $57.65\pm 0.14$ | $64.20\pm 0.02$ | $58.63\pm 0.12$ |

SP | $84.98\pm 0.16$ | $66.40\pm 0.31$ | $56.89\pm 0.71$ | $65.44\pm 0.04$ | $70.50\pm 0.13$ |

RW | $78.02\pm 0.20$ | $69.94\pm 0.27$ | $55.59\pm 0.01$ | $58.80\pm 0.04$ | $21.03\pm 0.22$ |

GR | $81.93\pm 0.17$ | $52.34\pm 0.42$ | $56.20\pm 0.10$ | $62.28\pm 0.02$ | $67.22\pm 0.11$ |

WL | $84.62\pm 0.23$ | $79.93\pm 0.35$ | $55.64\pm 0.20$ | $\mathbf{78.55}\pm \mathbf{0.04}$ | $31.33\pm 0.21$ |

**Table 3.**Classification accuracy (±standard error) on directed graph datasets, where ${}_{d}$ and ${}_{u}$ denote the directed and undirected versions of the datasets, respectively. The best and second best performing kernels are highlighted in bold and italic, respectively.

Kernel | ALZ${}_{\mathit{d}}$ | ALZ${}_{\mathit{u}}$ | SHOCK${}_{\mathit{d}}$ | SHOCK${}_{\mathit{u}}$ |
---|---|---|---|---|

QJSD${}_{A}$ | - | $\mathbf{65.87}\pm \mathbf{0.25}$ | - | $41.48\pm 0.15$ |

QJSD${}_{L}$ | $79.26\pm 0.24$ | $60.42\pm 0.23$ | $45.89\pm 0.23$ | $35.77\pm 0.21$ |

QJSD${}_{NL}$ | $\mathbf{82.07}\pm \mathbf{0.17}$ | $61.45\pm 0.22$ | $\mathbf{46.05}\pm \mathbf{0.20}$ | $\mathbf{44.38}\pm \mathbf{0.21}$ |

SP | $59.86\pm 0.25$ | $58.00\pm 0.29$ | $22.09\pm 0.29$ | $40.16\pm 0.24$ |

RW | $79.06\pm 0.21$ | $60.75\pm 0.25$ | $30.45\pm 0.26$ | $24.34\pm 0.28$ |

GR | $79.00\pm 0.20$ | $64.34\pm 0.27$ | $28.91\pm 0.30$ | $29.39\pm 0.28$ |

WL | $70.87\pm 0.27$ | $59.46\pm 0.35$ | $38.74\pm 0.27$ | $35.78\pm 0.26$ |

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**MDPI and ACS Style**

Minello, G.; Rossi, L.; Torsello, A. Can a Quantum Walk Tell Which Is Which?A Study of Quantum Walk-Based Graph Similarity. *Entropy* **2019**, *21*, 328.
https://doi.org/10.3390/e21030328

**AMA Style**

Minello G, Rossi L, Torsello A. Can a Quantum Walk Tell Which Is Which?A Study of Quantum Walk-Based Graph Similarity. *Entropy*. 2019; 21(3):328.
https://doi.org/10.3390/e21030328

**Chicago/Turabian Style**

Minello, Giorgia, Luca Rossi, and Andrea Torsello. 2019. "Can a Quantum Walk Tell Which Is Which?A Study of Quantum Walk-Based Graph Similarity" *Entropy* 21, no. 3: 328.
https://doi.org/10.3390/e21030328